The "battery saver" is just a battery disconnect, so when it is shut off you have no power at all for lights. It seems evident your battery(s) are weak. Have you chjecked the water in the cells lately? How old are they and what type?
It may be that your battery charge system isn't woprking, but you say the lights are bright when on shore power and that usually menas the cahrging system is operating as it should.
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL